I've searched all over this site but I haven't really found a satisfying answer to this problem.
I want to know a simple way to compute the differential (pushforward) of a smooth map such as $F:M_1 \times M_2 \to N_1 \times N_2$ between products of a manifolds. I'd also like to know about the special cases $F:M_1 \times M_2 \to N$ and $F:M \to N_1 \times N_2$. Here's what I have so far.
To get a map like $F:M \to N_1 \times N_2 $, I can construct it using an injection $i_{(1,q_0)}: N_1 \to N_1 \times N_2$ define by $ p \mapsto (p,q_0)$, and compose it with $f_1 : M \to N_1$ by $f_1 (p) := \pi_1 (F(p,x))$ and considering the composition $f_1 \circ i_{(1,q_0)}$ which maps from $M$ to $N_1 \times N_2$ and use the chain rule to compute the pushforward. I would define $i_{(2,p_0)}:N_2 \to N_1 \times N_2$ and $f_2: M \to N_2$, analogously. The problem is that the composition is definitely not surjective while $F$ certainly could be (consider most smooth mappings $F:\mathbb{R}^5 \to S^1 \times S^1$).
Does anyone have any ideas for how to proceed?
PS. I am doing this because I want to solve a particular problem in Loring Tu's Introduction to Manifolds, problem 15.9(a), which says to prove that $f: GL(n,\mathbb{R}) \to SL(n,\mathbb{R}) \times R^\times$ defined by $A \mapsto (AM_{1/{\text{det}A}} , \text{det}A)$ is a diffeomorphism with $M_r := [re_1,e_2,\cdots,e_n]$ where $e_1,\dots,e_n$ is the standard basis for $\mathbb{R}^n$. To solve this, I need to compute the pushforward for a map between products of manifolds.
Have you tried to prove that is a bijective map (is not hard) and then proving that is a local diffeomorphism, by deducing that the rank of the map is maximal (at this point apply Inverse Function Theorem to prove that is a local diffeomorphism). Bijective local diffeomorphism are diffemorphisms beacause differentiability is a local property.